Fixed-Node Diffusion Quantum Monte Carlo Method on Dissociation

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Fixed-Node Diffusion Quantum Monte Carlo Method on Dissociation Energies and Their Trends for R-X Bonds (R= Me, Et, i-Pr, t-Bu) Aiqiang Hou, Xiaojun Zhou, Ting Wang, and Fan Wang J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03149 • Publication Date (Web): 07 May 2018 Downloaded from http://pubs.acs.org on May 7, 2018

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The Journal of Physical Chemistry

Fixed-Node Diffusion Quantum Monte Carlo Method on Dissociation Energies and Their Trends for R-X Bonds (R= Me, Et, i-Pr, t-Bu)

Aiqiang Hou, Xiaojun Zhou, Ting Wang and Fan Wang*a)

Institute of Atomic and Molecular Physics, Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu, P. R. China

a) Corresponding author: [email protected] 1

ACS Paragon Plus Environment

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ABSTRACT Achieving both bond dissociation energies (BDEs) and their trends for the R-X bonds with R=Me, Et, i-Pr and t-Bu reliably is nontrivial. DFT methods with traditional exchange-correlation (XC) functionals usually have large error on both the BDEs and their trends. The M06-2X functional gives rises to reliable BDEs but the relative BDEs are determined not as accurately. More demanding approaches such as some double-hybrid functionals, G4 and CCSD(T) is generally required to achieve the BDEs and their trends reliably. The fixed-node diffusion quantum Monte Carlo method (FN-DMC) is employed to calculated BDEs of these R-X bonds with X=H, CH3, OCH3, OH and F in this work. The single-Slater-Jastrow wavefunction is adopted as trial wavefunction and PPs developed for QMC calculations are chosen. Error of these PPs is modest in wavefunction methods, while it is more pronounced in DFT calculations. Our results show that accuracy of BDEs with FN-DMC is similar to that of M06-2X and G4 and trends in BDEs are calculated more reliably than M06-2X. Both BDEs and trends in BDEs of these bonds are reproduced reasonably with FN-DMC. FN-DMC using PPs can thus be applied to BDEs and their trends of similar chemical bonds in larger molecules reliably and provide valuable information on properties of these molecules.

2


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I. INTRODUCTION Reliable prediction on bond dissociation energies (BDEs) as well as trends in BDEs of a same chemical bond in different chemical environment is highly important in computational chemistry to understand kinetics and thermodynamics of many chemical reactions. In addition, BDE is also closely related to relative stability of radicals. It has long been well-established in organic chemistry that relative stability of alky radicals is tertiary > second > primary > methyl.1 This trend of relative stability can be rationalized from hyperconjugation effect between the methyl groups and the radical center.2 BDEs of R-H and R-CH3 for R= Me, Et, i-Pr and t-Bu are consistent with relativity stability of the involved radicals and relative stability of these radicals are usually estimated from BDEs of R-H or R-CH3 bond.1,3 However, trends in BDE of R-X bonds are not necessarily consistent with relative stability of alky radicals for X other than H or CH3. Difference between electronegativity of R and X also plays an important role on BDE of a R-X bond.2 In fact BDEs of R-F or R-OH bonds increases from Me, Et and i-Pr to t-Bu, which is reversed to relative stability of these radicals. This order of BDEs in R-F and R-OH originates from large electronegativity of F and OH through an increased resonance between covalent and ionic form in R-X.2 For X with large electronegativity, contribution of the ionic configuration R+X- becomes more significant. R-X will thus be stabilized by the ionic configuration in the sequence of Me < Et < i-Pr < t-Bu. BDEs of R-X will thus increases from Me to t-Bu for X=F and OH.5 Furthermore, other effects such as steric strain between R and X, geometry relaxation could also affect BDEs of R-X.4-5 we focus on BDEs of R-X bonds and their trends with R=Me, Et, 3



i-Pr and t-Bu and X=H, CH3, OCH3, OH and F in this work. Density functional theory (DFT)6-7 is nowadays one of the most popular methods in electronic structure calculations due to its compromise between accuracy and efficiency. However, DFT results depend critically on the employed exchange-correlation (XC) functional. It has been shown2 that BDEs of R-X bonds are generally underestimated using the traditional XC functionals such as B3LYP8-10 and error of the obtained BDEs tends to become more pronounced for larger molecules.11 Furthermore, qualitative trends in BDEs of the R-X bond cannot be predicted correctly with many popular XC functionals for X with large electronegativity.2 BDEs of R-OH bonds are calculated to decrease from Me to t-Bu with these XC functionals, which is contrary to experimental trends.2 Poor behavior of these XC functionals on BDEs and trends in BDEs of the R-X bonds stems from overestimation on stability of Et, i-Pr and t-Bu compared with Me and this overestimation is more severe for i-Pr and t-Bu. It should be noted that BDEs of R-X bonds can be improved to some extent using an isodesmic method12 by referring BDE of the Me-X bond due to a systematic error cancellation.13However, trends in the BDEs will not change using the isodesmic method. On the other hand, some recently developed XC functionals such as the M06-2X14 functional and the double hybrid functional XYG311 can afford better BDEs and correct qualitative trends in BDEs of these R-X bonds. However, the M06-2X functional is heavily parameterized and computational effort of the double hybrid functional scales as N5, where N represents system size and it becomes costly for large systems. Wavefunction methods are more expensive to account for electron correlation 4


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compared with DFT. It has been shown that error on BDEs of R-X bonds is quite large with MP215 based methods, however qualitative trends in BDEs are reproduced correctly.2 Higher level wavefunction methods are required to achieve more reliable BDEs. The coupled-cluster approach (CC)16 at the singles and doubles level augmented by a perturbative treatment of triples (CCSD(T))17 is the so called “gold standard” in quantum chemistry18 and can capture dynamic correlation with high accuracy. However, computational effort of the CCSD(T) approach scales as N7 and a large basis set is generally needed to obtain highly accurate results. The CCSD(T) approach is thus applied mainly to rather small molecules. Some composite approaches such as G3and G4 approaches are developed to achieve reliable thermodynamic chemistry at a relatively low computational cost.19-20 BDEs of R-X bonds with G3 and G4 can be reproduced rather reliably. On the other hand, trends in BDEs of R-X with G3 are not very satisfactory, while results of G4 are much improved.11 However, CCSD(T) and a closely related QCISD(T)21 approach using a relatively small basis set is included in G4 and G3 calculations, respectively, and these approaches will still be quite demanding for large molecules. On the other hand, the quantum Monte Carlo (QMC) method22-25 developed in recent decades is a promising method in dealing with electron correlation. The diffusion Monte Carlo method (DMC)23 is the most popular QMC method in electronic structure calculations for molecules and solids and highly accurate results can usually be achieved. Importance sampling is used in DMC calculations to enhance efficiency through a trial wave function23 and the Fermion sign problem is fixed by confining the 5



configurations within node pockets of the trial wavefunction in most practical calculations, i.e. the fixed-node diffusion quantum Monte Carlo method (FN-DMC).26 Accuracy of FN-DMC results depend on location of the node surfaces of the trial wave function. Quality of the trial wave function thus decides both accuracy and efficiency of DMC calculations. The use of pseudopotentials27 (PP) is another common practice in DMC calculations and it can reduce computational cost by orders of magnitude. Due to its stochastic nature, exploitation of massively parallel computers is straightforward with DMC and it has been reported recently that an almost perfect parallel scaling is obtained for computations up to more than a hundred thousand cores.28 Computational effort of DMC scales as N2~3 with system size,24 but with a large prefactor. In addition, requirement on hard disk and memory of DMC calculations is also rather modest. This means DMC can possibly be applied to large systems with high accuracy at affordable computational resources, although it is still quite expensive for small molecules. DMC has been applied with great success to a large variety of problems such as BDEs,29 barrier heights,30-31 reaction energies,32 atomization energies33-34 and noncovalent interactions.25 Recent works on BDEs of O-H bond in phenol29 and C-H bond in acetylene35 shows that DMC can provide reliable BDEs with a proper trial wavefunction. In our previous work, BDEs of X-H involved in 19 hydrogen transfer reactions are calculated using FN-DMC.30 Mean unsigned error for those BDEs is 0.6kcal/mol with all-electron FN-DMC using the Slater-Jastrow trial wavefunction and agreement with reference values is even better than CCSD(T) with the aug-cc-pVQZ basis set.36 We 6


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propose to investigate performance of FN-DMC on BDEs of the above mentioned R-X bonds and their trends in this work. It should be noted that some of these BDEs are included in their benchmark database as the ABDE12 set37 in developing new XC functionals by the Truhlar group.

II. COMPUTATIONAL DETAILS

r The trial wavefuction ψ T ( x ) introduced in DMC calculations to exploit importance sampling usually takes the following form: r

r

r

ψ T ( x ) = ψ A ( x )e J ( x )

(1)

r r whereψ A ( x ) is an antisymmetrized wavefunction and J ( x ) is the Jastrow factor.38In r most DMC calculations, ψ A ( x ) is taken to be a Slater determinant obtained from a HF or DFT calculation. More complicated antisymmetrized wavefunction such as multi-determinant wavefunction is required to improve accuracy of DMC results.39-40 In addition, a multi-determinant wavefunction or wavefunctions suitable for description of non-dynamic correlation is necessary for systems with a significant multireference character. The molecules and their fragments investigated in this work can be described reasonably using single-reference methods. Moreover, computational effort of obtaining a suitable multi-determinant trial wavefunction for DMC calculations is usually quite demanding compared with a HF or DFT calculation. The orbitals required to construct the determinant are typically expanded in a basis set.24 The most commonly used basis sets include plane waves, Slater-type orbitals (STO), Gaussian-type orbitals (GTO), and numerical orbitals. The Gaussian basis set is 7



employed in this work. The Jastrow factor provides a reasonable description on dynamic correlation41 and trial wavefuntion of single-Slater-Jastrow form can usually afford rather accurate DMC results for systems with a dominant single-reference character. Trial wavefunctions of single-Slater-Jastrow form are thus employed in the present DMC calculations. It has been shown previously that DMC results using the determinant composed of DFT orbitals are improved compared with those using the HF determinant.33 Furthermore, DMC results are not sensitive to the employed exchange-correlation (XC) functional and the determinant wavefuction obtained from the simplest XC functional, the LDA functional42 is adopted in the present work. It has been shown in our previous work that DMC calculations using LDA orbitals can provide reliable barrier heights for H-abstraction reactions and the corresponding X-H BDEs.30 The Jastrow factor is employed to describe dynamic correlation and to enforce electron-electron cusp condition.38 The form of the Jastrow factor employed in this work is proposed in Ref(38). and is the sum of homogeneous, isotropic electron-electron terms, isotropic electron-nucleus terms centered on the nuclei, and isotropic electron-electron-nucleus terms.Node surfaces are not changed by the Jastrow factor, but efficiency, stability and time-step bias of DMC calculations are enhanced enormously. On the other hand, the Jastrow factor will also affect the FN-DMC energy when PPs are employed due to an approximate treatment of nonlocal terms in PPs.43 The parameters in the Jastrow factor thus need to be optimized carefully to achieve reliable DMC results when PP is adopted. The Jastrow factors are optimized using the 8


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energy minimization method44 together with the variance minimization method45 based on variational Monte Carlo (VMC)46 calculations. Different combinations of optimization steps are tested and we found that two energy minimization steps followed by one variance minimization step can usually provide an optimal Jastrow factor. 2×104 configurations are chosen from 2×106 configurations generated from VMC calculations for energy or variance minimization and 5 cycles of configuration generation and optimization are carried out in each of the three optimization steps. In our DMC calculations, 2048 configurations sampled according to square of the trial wavefunction is initially employed. These configurations propagate in real space according to a Green’s function which is decomposed to a drift-diffusion Green’s function and a branching Green’s function.23 This decomposition is exact only in the limit of zero time step and it results in the time-step bias of DMC results. To eliminate the time-step bias, DMC calculations are carried out at time steps of 0.02au, 0.06 au and 0.10au, respectively. A quadratic extrapolation is performed subsequently to obtain the final DMC energy at zero time step in our work. The configurations are restricted to move within the node pocket of the trial wavefunction in the fixed-node approximation. A total number of 6000 equilibrium steps are performed before the DMC statistics accumulation phase in the present DMC calculations. Variance of the total energy is obtained using the reblocking method47to reduce effects of serial correlation in the calculations. Kinetic and potential energies change rapidly when an electron moves near a bare nuclear. A rather small time-step has to be chosen and most of the computational effort 9



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is spent on averaging out large fluctuations of energies around nucleus. Computational cost of DMC calculations scales as Z5.5~6.5 where Z is the nuclear charge.48 Use of PPs can thus increase efficiency of DMC calculations by orders of magnitude. PPs take the following form in electronic structure calculations27:

VPP = −

Z eff r

+

lmax −1

l

∑ ∑

l = 0 m =− l

Ylm Vl − Vlmax  Ylm + Vlmax

Vl = ∑ Alk r nlk − 2 e − Blk r

(2)

2

(3)

k

where Zeff is the effective nuclear charge. In PPs developed in traditional quantum chemistry calculations, nlk is chosen to be 0, 1 or 2 and the potential diverges at nuclear when nlk is 0 or 1 and the PP does not suite for QMC calculations. On the other hand, nlk is set to be equal or larger than 2 in DMC calculations except that Vlmax , i.e. the local part of the PP, contains a term with Alk=Zeff and nlk=1. This term cancels out divergence of the -Zeff/r term at the nuclear position. Additional approximation is required to deal with the nonlocal terms in the PPs, i.e. the second term on the right hand side of Eq. (2). A “T-move” scheme49 is adopted in this work to deal with the nonlocal terms. Total energy obtained with this scheme satisfies the variational principle and DMC calculations are more stable compared with the local approximation. Detailed comparisons between T-move and local approximation are given in Ref.(50,51). PPs developed by Trail and Needs based on Dirac-Fock data (TNDF)52 as well as those by Burkatzki, Filippi, and Dolg based on a scalar relativistic HF theory (BFD)53 are employed in this work. The BFD potential for H atom is not used since it is not very accurate for this element. Many other PPs suitable for QMC calculations have been developed,54-57 however, the Gaussian basis sets corresponding to those PPS are usually 10


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not provided. The aug-cc-pVQZ basis set52 for the TNDF potential and the cc-pVQZ basis set53 for the BFD potential are chosen in obtaining the determinant wavefunction. It should be noted that DMC results are not very sensitive to the employed basis set.25 Besides DMC calculations, DFT calculations using the PBE58 B3LYP and M06-2X functionals as well as CCSD(T) calculations are also carried out with the aug-cc-pVQZ basis set.36 To evaluate reliability of the employed PPs, DFT and CCSD(T) calculations using the TNDF and BFD PPs and the corresponding basis sets are also performed and these results are compared with those of all-electron calculations. In addition, BDEs of the R-X bonds and their trend with G4 are also present. We are not able to perform CCSD(T) calculations for t-Bu-OCH3 using the aug-cc-pvQZ basis set due to its size and BDE of this bond with CCSD(T) is not provided. Geometries of the involved molecules or fragments are taken from those in the ABDE12 database of the Trular’s group37 except for i-Pr-OH, Me-OH and the fluorides, which are optimized using the B3LPY/6-31G(d)59 method. It should be noted that geometries in G4 are optimized using the B3LYP/6-31G(2df,p) 19 approach. DFT, CCSD(T) and G4 calculations are carried out using the Gaussian09 program package,60 and DMC calculations are performed with the CASINO program package.22 All calculations are performed on National Supercomputing Center of Shenzhen.

III. RESULTS AND DISCUSSIONS Error on PPs. Use of PPs reduces computational effort of DMC calculations pronouncedly, but it will inevitably introduce some errors. We will firstly evaluate 11



errors of the employed PPs on the BDEs. It should be noted that errors of the PPs are related to the employed electronic structure approach. All-electron DMC calculations for some of the involved molecules are quite expensive and we only compare the BDEs from all-electron calculations with those using the two PPs at DFT, HF, MP2, MP4, CCSD and CCSD(T) levels. Mean absolute deviation (MAD) of the BDEs using PPs from those of all-electron results are listed in Table 1 for the BFD and TNDF potentials. MAD for BDEs of the R-X bonds with the same X are also listed in this table. It can be seen from this tables that errors of the PPs on these BDEs are the smallest at the HF level. This is understandable since these PPs are developed from data of the HF method. In addition, errors of the PPs for ab initio correlated methods such as MP2 and CC approaches are slightly larger than those in the HF method, .e.g. 0.58kcal/mol and 0.70kcal/mol at the CCSD(T) level for the BFD and TNDF potential, respectively. On the other hand, error of the PPs is sizeable in DFT calculations. This indicates that these PPs are not applicable to DFT calculations. One can see from Table 1 that MAD of the BFD potential in correlated approaches is rather small for BDEs of the R-H and R-CH3 bonds. In fact, it is even smaller than that in HF calculations. On the other hand, MAD of the BFD potential reaches about 1kcal/mol for BDEs of the R-OH, R-OCH3 and R-F bonds with the correlated methods. As for BDEs using the TNDF potential, the MAD is the smallest for the R-H bonds. Similar to the case of the BFD potential, error on BDEs of R-OH, R-OCH3 and R-F bonds are larger than that of R-CH3 and R-H bonds using the TNDF potential. According to results in this table, error of the TNDF potential is slightly larger than that of BFD potential in correlated 12


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calculations. Accuracy of DMC methods is comparable to that of CCSD(T) and we expect that the mean absolute error of the PPs on these BDEs from DMC calculations should be smaller than 1kcal/mol. In addition, error of the PPs in DMC calculations will probably be larger for BDEs of R-OH, R-OCH3 and R-F bonds. BDEs of R-X Bonds. The obtained BDEs using DFT, CCSD, CCSD(T), G4 and DMC methods as well as experimental values61 are listed in the Supporting Information. Zero-point vibration energies obtained from B3LYP/6-31G(d) scaled by 0.9806 are included in experimental dissociation energies. MAD of the BDEs for the R-X bonds with the same X are listed in Table 2 together with MAD and mean deviation (MD) for BDEs of all the R-X bonds. Experimental values for BDEs of the Et-F, i-Pr-F and t-Bu-F bonds are not available and they are taken from the present all-electron CCSD(T)/aug-cc-pVQZ results, since BDEs of this approach agree best with experimental values of the other molecules. One can see from this table that BDEs of all-electron CCSD(T) agree best with the available experimental data and error of CCSD(T) results become slightly larger when PPs are used. BDEs of these bonds are also reproduced rather accurately with G4, M06-2X and DMC methods. It can be seen that error of DMC approaches on these BDEs is even smaller than that of CCSD(T) using cc-pVQZ basis when PPs are employed. As for DFT results, it is interesting to note that error of the PBE functional on these BDEs is actually smaller than that of the hybrid B3LYP functional. According to the MDs listed in this table, most of the employed approaches underestimate BDEs on average except for the DMC/BFD approach. In fact, MD of DMC results is rather close to the error bars and we cannot 13



draw the conclusion that the employed DMC approach tends to overestimate the BDEs. According to results in the Supporting Information, the present DMC approach overestimates BDEs of R-H and R-OCH3 bonds and underestimates BDEs of the R-OH and R-F bonds. BDEs of the R-OCH3 bond are also overestimated by all-electron CCSD(T) approach, while it underestimates BDEs of the other R-X bonds. It can be seen from this table that the M06-2X functional is able to provide rather accurate the BDEs except for the R-H bonds. As for the PBE and B3LYP functionals, although these two XC functionals cannot provide qualitatively correct trends on BDEs of the R-F and R-OH bonds, error on BDEs of these two types of bonds is not the largest. On the other hand, error on BDEs of R-H bonds is rather large among the R-X bonds for the PBE functional, although trends of BDEs of the R-H bond can be predicted correctly using the PBE functional. This indicates that an approach that can afford qualitatively correct trends in BDEs does not necessarily provide reasonable BDEs. It has been found2 that trends of the BDEs are reproduced reasonably with MP2, although errors of MP2 on BDEs of the R-X bonds are rather large. BDEs of the R-H, R-OCH3 and R-F bonds are predicted with high accuracy using the G4 method, while error for BDEs of R-CH3 and R-OH bonds are somewhat larger with G4. As for the CCSD(T)/aug-cc-pVQZ calculations, BDEs of R-OH bonds are underestimated by about 1kcal/mol, while highly accurate BDEs are achieved for the other R-X bonds. The present DMC calculations provide quite accurate BDEs for R-H and R-CH3 bonds, but errors on BDEs of R-OH and R-OCH3 bonds are somewhat larger. BDEs of the R-F bonds can be reproduced accurately in the present DMC calculations with the BFD 14


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potentials, and they are underestimated to some extent when the TNDF potential is used. It has been found that error of some popular XC functionals tends to be more pronounced for larger systems. We indeed found that error of the BDEs increases in the sequence of Me-X, Et-X, i-Pr-X and t-Bu-X for PBE and B3LYP. However, such trend cannot be observed for the other approaches including the M06-2X functional. Relative BDEs and Their Trends. MAD and MD on relative BDEs with respect to that of Me-X for all the R-X bonds using these approaches are listed in Table 3. MAD on relative BDEs for R-X bonds with the same X are also listed. One can see from this table that error of PBE and B3LYP on relative BDEs is rather large and this is consistent with previous findings2. MAD on relative BDEs using PBE is even larger than that with B3LYP, although BDEs with PBE agree better with experimental values than those of B3LYP. Relative BDEs of R-H bonds are calculated reliably using B3LYP, but they are quite poor for the other R-X bonds with this XC functional. On the other hand, PBE could not afford reasonable relative BDEs for all the R-X bonds. As shown in Table 2, M06-2X is able to provide BDEs accurately for these R-X bonds, but relative BDEs are not predicted as reasonably. According to results in this table, relative BDEs of R-H bonds are determined with high accuracy using M06-2X, while its performance deteriorates for the other R-X bonds. It can be seen from this table that all the employed DFT methods underestimate relative BDEs and this is because these DFT approaches overestimate stability of Et, i-Pr and t-Bu compared with Me. According to this table, relative BDEs are calculated rather accurately using G4 and all-electron CCSD(T). Error on relative BDEs of R-OCH3 bonds is more 15



pronounced than that of the other R-X bonds with G4, while relative BDEs for R-H bonds with CCSD(T) are not as accurate as those of the others. Relative BDEs are underestimated slightly with G4, and they are overestimated marginally using CCSD(T). On the other hand, DMC with both TNDF and BFD potentials provides rather reliable relative BDEs. This shows both BDEs and relative BDEs of these R-X bonds are reproduced reasonably with DMC. Compared with other approaches, error of relative BDEs with DMC is smaller than that with M06-2X and slightly larger than that of G4, although MADs for BDEs with G4, DMC and M06-2X are similar. Error on relative BDEs of R-H bonds is the largest with both DMC/TNDF and DMC/BFD followed by that of R-OCH3 bonds. On the other hand, relative BDEs of R-CH3 bonds estimated with DMC agree best with experimental data among all the R-X bonds. MD on relative BDEs with DMC is similar in magnitude to error bars of DMC calculations and one cannot conclude whether DMC overestimate or underestimate the relative BDEs on average. To further illustrate performance of these approaches on trends in BDEs, relative BDEs for these R-X bonds determined from PBE, B3LYP, M06-2X, G4, CCSD(T) and DMC calculations are plotted in Fig. 1. It can be seen from this figure that all the approaches correctly predict trends in BDEs of R-H bonds. Relative BDEs of R-H bond are underestimated significantly with PBE, while results of B3LYP agree rather well with experimental data except for the t-Bu-H bond. On the other hand, trend in BDEs of R-H bond with M06-2X agree best with experimental results among these approaches and the other methods tend to overestimate relative BDEs of R-H bonds to some extent. 16


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DMC results are quite close to CCSD(T) results for R-H bonds. As for R-CH3 bonds, all these approaches also provide qualitatively correct trends, although error of PBE and B3LYP is much more significant that the other approaches. M06-2X provides improved trend in BDEs of R-CH3 bonds compared with PBE and B3LYP, but its error is more pronounced than that of the other methods. Trends in BDEs of R-CH3 bonds with G4, CCSD(T) and DMC closely resemble experimental result. As for R-OCH3 bonds, relative BDEs cannot be predicted correctly with PBE and B3LYP. M06-2X provides qualitatively correct trends for these bonds, although relative BDEs are calculated to be too low. It is interesting to note that G4 underestimate relative BDE of i-Pr-OCH3 sizeably, while its results on the other R-OCH3 bonds are more reasonable. DMC overestimates relative BDEs of R-OCH3 bonds, but its results are better than those with G4. Performance on trend of R-OH bonds with PBE, B3LYP and M06-2X is similar to that of R-OCH3 bonds. G4, CCSD(T) and DMC/TNDF results are quite close to each other and these approaches underestimate relative BDEs of R-OH bonds on average. On the other hand, DMC/BFD tends to overestimate relative BDEs of R-OH bonds to some extent. As for R-F bonds, relative BDEs with M06-2X are again improved pronouncedly upon PBE and B3LYP, although they are still underestimated to some extent compared with CCSD(T) results. Relative BDE of Et-F are calculated too low with DMC/TNDF, while DMC/BFD results agree quite well with CCSD(T) results.

IV. CONCLUSION 17



Reliable BDEs of these R-X bonds are closely related to understanding and studying stabilities, reaction energies as well as reaction barriers of the involved species. In addition, reasonable trends in BDEs of these bonds are also important in rationalizing effects of substituents. However, it turns out to be non-trivial to achieve both reasonable BDEs and its trends for these R-X bonds. Error on BDEs with popular XC functionals such as PBE and B3LYP are usually quite large and trends in BDEs for R-X bonds of X with large electronegativity cannot be predicted even qualitatively. BDEs are improved pronounced with M06-2X, but the trends are not determined as accurately. In fact, some composite methods such as G3 could not provide reliable trends either. More demanding methods such as the double hybrid XC function XYG3, G4 and CCSD(T) can afford both BDEs and their trends with high accuracy. In this work we evaluate performance of FN-DMC on BDEs and their trends using the single-Slater-Jastrow trial wavefunction. PPs are usually adopted in FN-DMC to increase computational efficiency. The PPs employed in traditional quantum chemistry calculations cannot be adopted in DMC calculations and specially developed PPs that are free from singularity at nuclear is adopted. Error of these PPs is modest when wavefunction methods are used, while their error is pronounced in DFT calculations. Error of FN-DMC results on BDEs are comparable to those of M06-2X and G4, but FN-DMC outperforms M06-2X in trends of BDEs. MADs on BDEs using FN-DMC with TNDF and BFD potentials are 0.89/kcal and 0.75kcal/mol, respectively and MADs on relative BDEs are 0.61kcal/mol and 0.58kcal/mol. This shows both BDEs and trends in BDEs can be reproduced with high accuracy using FN-DMC. Due to its 18


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low computational scaling and high parallel efficiency, FN-DMC using the single-Slater-Jastrow and PPs can thus be applied reliably to investigation of BDEs and their trends of similar chemical bonds in larger molecules and provide valuable information on kinetic and thermodynamic properties of these molecules.

ACKNOWLEDGEMENT We thank the National Nature Science Foundation of China (grant no. 21473116 and 21773160) for financial support.

Supporting Information. In this section, we presented the Bond dissociation energies of R-X(X=H, CH3, OCH3, OH and F )bonds with R=Me, Et, i-Pr and t-Bu as well as MAD and MD for BDEs using different methods. REFERENCE

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Table 1. Mean absolute deviation (MAD) of bond dissociation energies using the BFD potential or the TNDF potential compared with those in all-electron calculations. (unit: kcal/mol) R-CH3 BFD potential HF 0.20 MP2 0.04 MP4 0.04 CCSD 0.03 CCSD(T) 0.07 PBE 3.86 B3LYP 3.36 M06-2x 3.34 TNDF potential HF 0.44 MP2 0.60 MP4 0.55 CCSD 0.60 CCSD(T) 0.66 PBE 4.81 B3LYP 4.25 M06-2x 1.86

R-H

R-OH

R-OCH3 R-F

all

0.11 0.08 0.09 0.06 0.03 1.87 1.46 1.60

0.42 1.04 0.98 1.06 1.06 4.45 4.10 4.47

0.50 1.06 0.96 1.01 0.98 3.56 3.69 4.01

0.53 0.92 0.81 0.81 0.77 4.92 4.54 4.15

0.35 0.63 0.58 0.59 0.58 3.73 3.43 3.51

0.22 0.25 0.21 0.23 0.26 2.45 1.82 1.16

0.71 0.93 0.81 0.81 0.85 5.54 5.05 2.85

0.64 0.86 0.74 0.74 0.78 4.51 4.47 3.05

0.85 1.08 0.93 0.92 0.94 6.35 5.67 2.47

0.57 0.74 0.65 0.66 0.70 4.73 4.25 2.28

27



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Table 2. MAD of bond dissociation energies with respect to experimental values for different R-X bonds and MAD and MD for all the R-X bonds. (unit: kcal/mol) PBE B3LYP M06-2X G4 CCSD CCSD/TNDF CCSD/BFD CCSD(T) CCSD(T)/TNDF CCSD(T)/BFD DMC/TNDF DMC/BFD

R-H

R-CH3

R-OH

R-OCH3

R-F

MAD

MD

4.60 3.34 1.26 0.60 1.25 1.48 0.91 0.46 0.64 0.44 0.58 0.66

3.77 8.72 0.58 1.08 3.50 4.11 3.54 0.53 1.22 0.64 0.43 0.45

2.47 7.81 0.66 1.30 4.55 5.36 5.61 1.03 1.87 2.08 1.46 0.92

6.14 10.12 0.85 0.77 3.07 3.81 4.08 0.38 0.40 0.60 0.95 1.15

2.60 4.17 0.70 0.49 3.58 4.50 4.39 ........ 0.97 0.8 1.04 0.58

3.96 6.83 0.81 0.85 3.19 3.85 3.70 0.60 1.02 0.91 0.89 0.75

-2.81 -6.83 -0.55 0.75 -3.19 -3.85 -3.76 -0.37 -1.00 -0.88 -0.35 0.10

28


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Table 3. MAD of relative BDEs with respect to experimental values for different R-X bonds and MAD and MD for all the R-X bonds. (unit: kcal/mol) PBE B3LYP M06-2X G4 CCSD(T) DMC/TNDF DMC/BFD

R-H

R-CH3

3.35 0.76 0.14 0.25 0.96 1.27 0.89

4.89 4.27 1.20 0.12 0.13 0.27 0.21

R-OH 4.39 3.90 1.70 0.46 0.47 0.42 0.60

R-OCH3 4.74 4.99 1.19 1.02 0.25 0.67 0.74

29


R-F 3.80 2.84 1.17 0.47 0.12 0.40 0.44

MAD 4.23 3.35 1.08 0.46 0.39 0.61 0.58

MD -4.24 -3.30 -1.08 -0.20 0.23 0.30 0.09


Fig. 1 Trends in bond dissociation energies for R-X bonds with different approaches.

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Fixed-Node Diffusion Quantum Monte Carlo Method on Dissociation

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